login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A301830 Number of factorizations of n into factors (greater than 1) of two kinds. 19
1, 2, 2, 5, 2, 6, 2, 10, 5, 6, 2, 16, 2, 6, 6, 20, 2, 16, 2, 16, 6, 6, 2, 36, 5, 6, 10, 16, 2, 22, 2, 36, 6, 6, 6, 46, 2, 6, 6, 36, 2, 22, 2, 16, 16, 6, 2, 76, 5, 16, 6, 16, 2, 36, 6, 36, 6, 6, 2, 64, 2, 6, 16, 65, 6, 22, 2, 16, 6, 22, 2, 108, 2, 6, 16, 16, 6 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

a(n) depends only on the prime signature of n. - Andrew Howroyd, Nov 18 2018

LINKS

Andrew Howroyd, Table of n, a(n) for n = 1..10000

Jacob Sprittulla, On Colored Factorizations, arXiv:2008.09984 [math.CO], 2020.

Index to sequences related to prime signature

FORMULA

Dirichlet g.f.: Product_{n > 1} 1/(1 - n^(-s))^2. [corrected by Ilya Gutkovskiy, Dec 14 2020]

a(p^n) = A000712(n) for prime p. - Andrew Howroyd, Nov 18 2018

EXAMPLE

The a(6) = 6 factorizations: (2*3)*(), (3)*(2), (2)*(3), ()*(2*3), (6)*(), ()*(6).

The a(12) = 16 factorizations:

  ()*(2*2*3), (2)*(2*3), (3)*(2*2), (2*2)*(3), (2*3)*(2), (2*2*3)*(),

  ()*(2*6), (2)*(6), (6)*(2), (2*6)*(), ()*(3*4), (3)*(4), (4)*(3), (3*4)*(),

  ()*(12), (12)*().

MATHEMATICA

facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];

Table[Sum[Length[facs[d]]*Length[facs[n/d]], {d, Divisors[n]}], {n, 100}]

PROG

(PARI) MultEulerT(u)={my(v=vector(#u)); v[1]=1; for(k=2, #u, forstep(j=#v\k*k, k, -k, my(i=j, e=0); while(i%k==0, i/=k; e++; v[j]+=binomial(e+u[k]-1, e)*v[i]))); v}

seq(n)={MultEulerT(vector(n, i, 2))} \\ Andrew Howroyd, Nov 18 2018

CROSSREFS

Cf. A000712, A001055, A001222, A001405, A122768, A276024, A281113, A284640, A295632, A299701, A299702, A299729, A301829.

Sequence in context: A240081 A305791 A299764 * A305799 A294339 A185291

Adjacent sequences:  A301827 A301828 A301829 * A301831 A301832 A301833

KEYWORD

nonn

AUTHOR

Gus Wiseman, Mar 27 2018

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified January 27 16:41 EST 2022. Contains 350611 sequences. (Running on oeis4.)